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Discontinuous Galerkin Methods for the Biharmonic Problem on Polygonal and Polyhedral Meshes

Discontinuous Galerkin Methods for the Biharmonic Problem on Polygonal and Polyhedral Meshes

Year:    2019

Author:    Zhaonan Dong

International Journal of Numerical Analysis and Modeling, Vol. 16 (2019), Iss. 5 : pp. 825–846

Abstract

We introduce an $hp$-version symmetric interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the biharmonic equation on general computational meshes consisting of polygonal/polyhedral (polytopic) elements. In particular, the stability and $hp$-version a-priori error bound are derived based on the specific choice of the interior penalty parameters which allows for edges/faces degeneration. Furthermore, by deriving a new inverse inequality for a special class of polynomial functions (harmonic polynomials), the proposed DGFEM is proven to be stable to incorporate very general polygonal/polyhedral elements with an $arbitrary$ number of faces for polynomial basis with degree $p$ = 2, 3. The key feature of the proposed method is that it employs elemental polynomial bases of total degree $\mathcal{P}$$p$, defined in the physical coordinate system, without requiring the mapping from a given reference or canonical frame. A series of numerical experiments are presented to demonstrate the performance of the proposed DGFEM on general polygonal/polyhedral meshes.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/2019-IJNAM-13256

International Journal of Numerical Analysis and Modeling, Vol. 16 (2019), Iss. 5 : pp. 825–846

Published online:    2019-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    22

Keywords:    Discontinuous Galerkin polygonal/polyhedral elements inverse estimates biharmonic problems.

Author Details

Zhaonan Dong